Comment on ’A uniform boundedness theorem for locally convex cones’ [W. Roth, Proc. Amer. Math. Soc. 126 (1998), 1973-1982] - Saeedi, D.; Nikoufar, I.; Saiflu, H. (Methods of Functional Analysis and Topology, 2014)
In page 1975 of [W. Roth, A uniform boundedness theorem for locally convex cones, Proc. Amer. Math. Soc. 126 (1998), no.7, 1973-1982] we can see: In a locally convex vector space E a barrel is defined to be an absolutely convex closed and absorbing subset A of E. The set U = {(a,b)E^2, a-bA} then is seen to be a barrel in the sense of Roth’s definition. With a counterexample, we show that it is not enough for U to be a barrel in the sense of Roth’s definition. Then we correct this error with providing its converse and an application.